Adjusting a Grandfather Clock's Pendulum for Accurate Timekeeping

Accurate timekeeping is crucial for both traditional and modern applications. In particular, the grandfather clock's pendulum plays a significant role in maintaining the clock's precision. If a 0.99-meter long pendulum causes the clock to run late by 1 minute each day, how can we adjust it to ensure the clock runs perfectly?

Understanding the Pendulum's Contribution

The grandfather clock's pendulum oscillates back and forth with a specific period. One day has 1,440 minutes, so a slight adjustment in the pendulum's period will affect the clock's accuracy. If the clock runs slow by 1 minute per day, this means the period of the pendulum is 1/1440 too long compared to the desired period.

Let's denote the desired period as ( T_d ) and the current period as ( T_c ). Given that the current period is too long by 1/1440, we can express it as:

[ T_c T_d frac{1}{1440} ]

The relationship between the period of the pendulum and the pendulum length is given by the formula ( T approx 2pi sqrt{frac{L}{g}} ), where ( T ) is the period (in seconds), ( L ) is the length of the pendulum (in meters), and ( g ) is the acceleration due to gravity (approximately (9.8 , text{m/s}^2)).

Calculating the Required Pendulum Length

To adjust the clock to run on time, we need to shorten the pendulum. The desired period ( T_d ) should be 0.9993056 of the current period ( T_c ). The new length ( L_d ) required to achieve this period can be calculated as:

[ L_d L_c left(frac{T_d}{T_c}right)^2 ]

Substituting the values, we get:

[ L_d 0.99 left(frac{0.9993056}{1 frac{1}{1440}}right)^2 ]

Calculating this expression step-by-step:

[ T_c 0.99 times left(1 frac{1}{1440}right) 0.99 times frac{1441}{1440} approx 0.9993056 ] [ left(frac{T_d}{T_c}right)^2 left(frac{0.9993056}{0.9993056}right)^2 frac{1439^2}{1440^2} ] [ L_d 0.99 times frac{1439^2}{1440^2} approx 0.99 times 0.998612 approx 0.988612 , text{m} ]

Converting this length to millimeters (since 1 meter 1000 millimeters), we get:

[ L_d approx 0.988612 times 1000 988.612 , text{mm} ]

Thus, the pendulum must be shortened by approximately 1.375 millimeters from its original length of 990 millimeters.

Conclusion

In summary, to adjust a 0.99-meter long pendulum for a grandfather clock that is running late by 1 minute per day, you need to shorten the pendulum by about 1.375 millimeters. This precise modification will help ensure the clock runs accurately.

If you need to accelerate the clock slightly, you can use the formula ( L - L' L frac{1439}{1440^2} ), which results in a pendulum that is about 1.37 millimeters shorter.